The first exotic spheres were constructed by John Milnor (1956) in dimension n = 7 as S3-bundles over S4. He showed that there are at least 7 differentiable structures on the 7-sphere. In any dimension Milnor (1959) showed that the diffeomorphism classes of oriented exotic spheres form the non-trivial elements of an abelian monoid under connected sum, which is a finiteabelian group if the dimension is not 4. The classification of exotic spheres by Michel Kervaire and John Milnor (1963) showed that the oriented exotic 7-spheres are the non-trivial elements of a cyclic group of order 28 under the operation of connected sum.

The unit n-sphere, Sn, is the set of all (n+1)-tuples(x1, x2, ... xn+1) of real numbers, such that the sum x12 + x22 + ... + xn+12 = 1. (S1 is a circle; S2 is the surface of an ordinary ball of radius one in 3 dimensions.) Topologists consider a space, X, to be an n-sphere if every point in X can be assigned to exactly one point in the unit n-sphere in a continuous way, which means that sufficiently nearby points in X get assigned to nearby points in Sn and vice versa. For example, a point x on an n-sphere of radius r can be matched with a point on the unit n-sphere by adjusting its distance from the origin by 1/r.

In differential topology, a more stringent condition is added, that the functions matching points in X with points in Sn should be smooth, that is they should have derivatives of all orders everywhere. To calculate derivatives, one needs to have local coordinate systems defined consistently in X. Mathematicians were surprised in 1956 when John Milnor showed that consistent coordinate systems could be set up on the 7-sphere in two different ways that were equivalent in the continuous sense, but not in the differentiable sense. Milnor and others set about trying to discover how many such exotic spheres could exist in each dimension and to understand how they relate to each other. No exotic structures are possible on the 1-, 2-, 3-, 5-, 6- or 12-spheres. Some higher-dimensional spheres have only two possible differentiable structures, others have thousands. Whether exotic 4-spheres exist, and if so how many, is an important unsolved problem in mathematics.

The monoid of smooth structures on n-spheres is the collection of oriented smooth n-manifolds which are homeomorphic to the n-sphere, taken up to orientation-preserving diffeomorphism. The monoid operation is the connected sum. Provided n ≠ 4, this monoid is a group and is isomorphic to the group Θn of h-cobordism classes of oriented homotopy n-spheres, which is finite and abelian. In dimension 4 almost nothing is known about the monoid of smooth spheres, beyond the facts that it is finite or countably infinite, and abelian, though it is suspected to be infinite; see the section on Gluck twists. All homotopy n-spheres are homeomorphic to the n-sphere by the generalized Poincaré conjecture, proved by Stephen Smale in dimensions bigger than 4, Michael Freedman in dimension 4, and Grigori Perelman in dimension 3. In dimension 3, Edwin E. Moise proved that every topological manifold has an essentially unique smooth structure (see Moise's theorem), so the monoid of smooth structures on the 3-sphere is trivial.

The group bPn+1 is a cyclic group, and is trivial or order 2 except in case n = 4k+3, in which case it can be large, with its order related to the Bernoulli numbers. It is trivial if n is even. If n is 1 mod 4 it has order 1 or 2; in particular it has order 1 if n is 1, 5, 13, 29, or 61, and Browder (1969) proved that it has order 2 if n = 1 mod 4 is not of the form 2k – 3. It follows from the now almost completely resolved Kervaire invariant problem that it has order 2 for all n bigger than 125; the case n = 125 is still open. The order of bP4k for k ≥ 2 is

22k−2(22k−1−1)B{\displaystyle 2^{2k-2}(2^{2k-1}-1)B\,\!}

where B is the numerator of | 4B2k/k |, and B2k is a Bernoulli number. (The formula in the topological literature differs slightly because topologists use a different convention for naming Bernoulli numbers; this article uses the number theorists' convention.)

where πnS is the nth stable homotopy group of spheres, and J is the image of the J-homomorphism. As with bPn+1, the image of J is a cyclic group, and is trivial or order 2 except in case n=4k+3,{\displaystyle n=4k+3,} in which case it can be large, with its order related to the Bernoulli numbers. The quotient group πnS/J{\displaystyle \pi _{n}^{S}/J\,} is the "hard" part of the stable homotopy groups of spheres, and accordingly Θn/bPn+1{\displaystyle \Theta _{n}/bP_{n+1}} is the hard part of the exotic spheres, but almost completely reduces to computing homotopy groups of spheres. The map is either an isomorphism (the image is the whole group), or an injective map with index 2. The latter is the case if and only if there exists an n-dimensional framed manifold with Kervaire invariant 1, which is known as the Kervaire invariant problem. Thus a factor of 2 in the classification of exotic spheres depends on the Kervaire invariant problem.

As of 2012[update], the Kervaire invariant problem is almost completely solved, with only the case n = 126 remaining open; see that article for details. This is primarily the work of Browder (1969), which proved that such manifolds only existed in dimension n = 2j − 2, and Hill, Hopkins & Ravenel (2009), which proved that there were no such manifolds for dimension 254 = 28 − 2 and above. Manifolds with Kervaire invariant 1 have been constructed in dimension 2, 6, 14, 30, and 62, but dimension 126 is open, with no manifold being either constructed or disproven.

The order of the group Θn is given in this table (sequence A001676 in the OEIS) from (Kervaire & Milnor 1963) (except that the entry for n = 19 is wrong by a factor of 2 in their paper; see the correction in volume III p. 97 of Milnor's collected works).

When I came upon such an example in the mid-50s, I was very puzzled and didn't know what to make of it. At first, I thought I'd found a counterexample to the generalized Poincaré conjecture in dimension seven. But careful study showed that the manifold really was homeomorphic to S7. Thus, there exists a differentiable structure on S7 not diffeomorphic to the standard one.

One of the first examples of an exotic sphere found by Milnor (1956, section 3) was the following: Take two copies of B4×S3, each with boundaryS3×S3, and glue them together by identifying (a,b) in the boundary with (a, a2ba−1), (where we identify each S3 with the group of unit quaternions). The resulting manifold has a natural smooth structure and is homeomorphic to S7, but is not diffeomorphic to S7. Milnor showed that it is not the boundary of any smooth 8-manifold with vanishing 4th Betti number, and has no orientation-reversing diffeomorphism to itself; either of these properties implies that it is not a standard 7-sphere. Milnor showed that this manifold has a Morse function with just two critical points, both non-degenerate, which implies that it is topologically a sphere.

Given an (orientation-preserving) diffeomorphism f : Sn−1 → Sn−1, gluing the boundaries of two copies of the standard disk Dn together by f yields a manifold called a twisted sphere (with twistf). It is homotopy equivalent to the standard n-sphere because the gluing map is homotopic to the identity (being an orientation-preserving diffeomorphism, hence degree 1), but not in general diffeomorphic to the standard sphere. (Milnor 1959b) Setting Γn{\displaystyle \Gamma _{n}} to be the group of twisted n-spheres (under connect sum), one obtains the exact sequence

For n > 5, every exotic n-sphere is diffeomorphic to a twisted sphere, a result proven by Stephen Smale which can be seen as a consequence of the h-cobordism theorem. (In contrast, in the piecewise linear setting the left-most map is onto via radial extension: every piecewise-linear-twisted sphere is standard.) The group Γn of twisted spheres is always isomorphic to the group Θn. The notations are different because it was not known at first that they were the same for n = 3 or 4; for example, the case n = 3 is equivalent to the Poincaré conjecture.

If M is a piecewise linear manifold then the problem of finding the compatible smooth structures on M depends on knowledge of the groups Γk = Θk. More precisely, the obstructions to the existence of any smooth structure lie in the groups Hk+1(M, Γk) for various values of k, while if such a smooth structure exists then all such smooth structures can be classified using the groups Hk(M, Γk). In particular the groups Γk vanish if k < 7, so all PL manifolds of dimension at most 7 have a smooth structure, which is essentially unique if the manifold has dimension at most 6.

In 4 dimensions it is not known whether there are any exotic smooth structures on the 4-sphere. The statement that they do not exist is known as the "smooth Poincaré conjecture", and is discussed by Michael Freedman, Robert Gompf, and Scott Morrison et al. (2010) who say that it is believed to be false.

Some candidates for exotic 4-spheres are given by Gluck twists (Gluck 1962). These are constructed by cutting out a tubular neighborhood of a 2-sphere S in S4 and gluing it back in using a diffeomorphism of its boundary S2×S1. The result is always homeomorphic to S4. But in most cases it is unknown whether or not the result is diffeomorphic to S4. (If the 2-sphere is unknotted, or given by spinning a knot in the 3-sphere, then the Gluck twist is known to be diffeomorphic to S4, but there are plenty of other ways to knot a 2-sphere in S4.)

Akbulut (2009) showed that a certain family of candidates for 4-dimensional exotic spheres constructed by Cappell and Shaneson are in fact standard.